PROGRAM HELPS QUICKLY CALCULATE DEVIATED WELL PATH

Mark P. Gardner
Plains Resources Inc.
Houston

A Basic computer program quickly calculates the angle and measured depth of a simple directional well given only the true vertical depth and total displacement of the target. Many petroleum engineers and geologists need a quick, easy method to calculate the angle and measured depth necessary to reach a target in a proposed deviated well bore. Too many of the existing programs are large and require much input data.

The drilling literature is full of equations and methods to calculate the course of well paths from surveys taken after a well is drilled. Very little information, however, covers how to calculate well bore trajectories for proposed wells from limited data. Furthermore, many of the equations are quite complex and difficult to use.

Fig. 1 lists a computer program with the equations to calculate the well bore trajectory necessary to reach a given displacement and true vertical depth (TVD) for a simple build plan. It can be run on an IBM compatible computer with MS-DOS version 5 or higher, QBasic, or any Basic that does not require line numbers. QBasic 4.5 compiler will also run the program. The equations are based on conventional geometry and trigonometry.

EQUATIONS

The equations are derived from the well bore distances and angles shown in Fig. 2. The variables used in the equations are shown in Fig. 2. The following subscripts are used with these variables: T = total, B = build, R = run, KOP = kick off point, and _ = delta. Equations 1-3 are simple geometric equations, and Equations 4-6 are based on standard trigonometry.

The displacement and true vertical depth are then found by using geometry and trigonometry with these equations (Equations 7 and 8). With the starting angle set to zero (01 = 0, and cos O = 1), Equation 7 simplifies into Equation 9.

Substituting the build displacement, DispB, into Equation 1 yields Equation 10, which is then solved for the run displacement, DispR (Equation 11).

Equation 12 is derived from Equation 3 and the true vertical depth of the run (Equation 6), TVDR and the true vertical depth of the build (Equation 8), TVDB-DispR from Equation 11 is then substituted in Equation 12 to yield Equation 13.

Both sides of Equation 13 are multiplied by tan 0 and rearranged to yield Equation 14, which can be simplified into Equation 15. Because Cos2 0 + sin2 0 = 1, Equation 15 can be further simplified into Equation 16.

ITERATION

Equation 16 cannot be solved by further reduction; the solution is obtained through iteration. The equation can be solved by rearranging terms and iterating the angle 0 until Deltaerr is within acceptable tolerances, or as close to zero as practical (Equation 17). Engineering judgment must be used to decide the acceptable tolerances for each well.

Because this equation requires an iterative solution, the first guess should be assumed to be between 90 (upper boundary) and 0 (lower boundary). The next iteration should use an angle midway between the first guess and one of the boundaries. By using the midpoint between each guess and a boundary, the function diverges quickly. Therefore, it is not necessary to take the derivative of the function.

The following notes are important for running the program:

• The program's input angle must be in radians.

• Because the run, R, is conventionally given in degrees/100 ft in the oil field, this value must be converted to radians/ft (1 = p/180 radians).

• A first guess for the angle 0 can be estimated by ignoring the build section and taking the arctangent of the total displacement, DispT, divided by the change in true vertical depth, TVD_.

• Note that the top and bottom boundaries change depending on the result of each guess for the iteration.

Copyright 1993 Oil & Gas Journal. All Rights Reserved.